I was wondering the same thing as the OP, and stumbled upon this thread when I searched. From the book
jhayes_tucson
suggested, they give the formula for the *photon rate at a focal plane from a star of magnitude m* (which I post here under the doctrine of Fair Use for the purpose of commentary and research):

(1) *S* = *NT*Ï€/4(1-Îµ^{2})*D*^{2} Î”Î»10^{-0.4m}

where:

*S* is the photon flux in photons/second
*N* is the irradiance of a magnitude-zero reference star
*T*, Îµ, *D*, and Î”Î» are parameters related to the telescope, atmospheric transmittance, and bandpass
*m* is the magnitude

Instead of guessing/researching a bunch of parameters to use the above equation, one could use their own telescope/camera to establish a simple relationship between magnitude and electron flux.

The photon flux can be expressed as the electron flux, *I*, divided by the quantum efficiency *Î·, *so:

(2) *I* / Î· = *NT*Ï€/4(1-Îµ^{2})*D*^{2} Î”Î»10^{-0.4m}

where *I* is in electrons/second. Rearranging to isolate *I*:

(3) *I* = Î·*NT*Ï€/4(1-Îµ^{2})D^{2} Î”Î»10^{-0.4m}

If we lump the quantum efficiency, reference star irradiance, transmittance, telescope, and bandpass parameters into a single constant *K*, we get a simple relationship between magnitude and electron flux:

**(4) ***I* = *K* 10^{-0.4m}

If one uses their own telescope/camera to measure the electron flux of an object with known magnitude, *m*_{test}, they can calculate the constant *K* for their specific optical/imaging train for a certain degree of atmospheric extinction. For example, I could point my telescope near the zenith away from extended objects, and measure the sky fog with a Sky Quality Meter. The electron flux of the sky could be calculated by using the mean value of a test exposure, adjusted for bias and dark current.

So, for the test measurement, we have:

(5) *I*_{test} = *K* 10^{-0.4mtest}

rearranging gives:

**(6) ***K* = *I*_{test} / 10^{-0.4mtest}

Once we've calculated *K*, we can thereafter use equation (4) to estimate the electron flux for a different object at the same atmospheric extinction with known magnitude *m*.

To adjust for atmospheric extinction, you can reduce *m* by 0.2 magnitudes per airmass, or use whatever other extinction model you see fit to adjust *m*.

Atmospheric transparency varies from night-to-night, so I assume the electron flux estimate from equation (4) would only be a rough estimate if *K* was calculated on a different night. I suppose one could calculate *K* for nights of different transparency, and use the one that most closely matches the night when one wants to estimate the electron flux for an object of known magnitude.

The formula given by the book is for the entire focal plane, but I'm pretty sure the relationship that I derived at equation (4) applies at the per-pixel level.

**Disclaimer**: I'm still fairly new at AP, so it's quite possible I made a mistake in my reasoning above. I'd be happy to hear critiques or corrections.

**Edited by ecorm, 23 March 2019 - 04:33 PM.**